I don't really see why I should need to say any more since I've already
given you all the information you need to see for yourself that your data
is consistent with a wide range of values for n (and, therefore, does not
determine the value of n with any accuracy.) Nevertheless, here is some
of why your analysis is suspect:
Your analysis is critically dependent upon associating specific values of
v with specific values of a. This is more than a little problematic
because 1) both v and a are subject to significant errors due to
differencing and double differencing data values which don't differ by a
lot more than their inherent uncertainties in the first place and 2) both
v and a are changing with an a priori *unknown* dependence on time.
Therefore, in order to obtain the required correlations (of specific
values of v with specific values of a, you have to 1) do a lot of data
smoothing and 2) decide how you are going to get values of v and a at the
*same* times. Both of these operations depend *critically* on your
assumptions about how the values *vary* with time ... which, again, is
unknown.
For instance, if you decide to associate the average velocity calculated
for each time interval with the midpoint of the interval, you are making
the implicit assumption that the acceleration is constant over that
interval. If the drag is velocity dependent, however, this is clearly
*not* the case, so following this procedure would introduce an error (not
just more uncertainty) into the analysis.
Similar considerations apply to the choice of the time with which to
associate the calculated accelerations, but here the situation is further
complicated by the gross errors introduced by the double differencing. Now
there are *lots* of choices about how to do the smoothing/averaging
and each one carries implicit assumptions about how the acceleration
varies with time ... again, something you don't know. If, for instance,
you do a strict average and associate the result with the midpoint of the
interval you are making the implicit assumption that the jerk rate is
constant which is, again, simply not the case for *any* reasonable drag
force.
The best way to avoid all these thorny difficulties is to use the model to
fit the measured values of d directly as I have done. Again, when I do so
with your data, I get good fits (with small rms deviations between model
d's and observed d's and no obvious trends in the residuals) over *large*
ranges of n.
.... and now I really must sign off on this topic and get to work grading
the mounds of final exams and late labs on my desk!
John
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A. John Mallinckrodt http://www.intranet.csupomona.edu/~ajm
Professor of Physics mailto:ajmallinckro@csupomona.edu
Physics Department voice:909-869-4054
Cal Poly Pomona fax:909-869-5090
Pomona, CA 91768-4031 office:Building 8, Room 223